Today we covered testing \(\hat{\beta_0}~and~\hat{\beta_1}\), Confidence Intervals for\(\hat{\beta_0}~and~\hat{\beta_0}\), Prediction vs Confidence Intervals, variation of response, Ftests.
First we went over testing \(\hat{\beta_0}~and~\hat{\beta_1}\). The test stat for \(\hat{\beta_1}\) is: \(\dfrac{\hat{\beta_1}}{StandardError( \hat{\beta_1})}\). It has a standard normal distribution.
The test stat for \(\hat{\beta_0}\) is the same, just replace each \(\hat{\beta_1}\) with \(\hat{\beta_0}\).
Then we talked about confidence intervals for \(\hat{\beta_0}\) and \(\hat{\beta_1}\), which are again the same. I’m just going to type out the CI for \(\hat{\beta_0}\) and you can go from there for \(\hat{\beta_1}\): CI for \(\hat{\beta_0}\): [\(\hat{\beta_0} \pm t^{(n-2)}_{(alpha/2)} * SE(\hat{\beta_0})\)]
Then we discussed confidence intervals for mean value of y when x = \[{x_0}\] and prediction intervals for y when x is \({x_0}\). The confidence interval is:
\[[~\hat{y} \pm t^{n-2}_{\alpha/2}*\sqrt{MSE}*\sqrt{Distance Value}~]\].
The Prediction interval is:
\[[~\hat{y} \pm t^{n-2}_{\alpha/2}*\sqrt{MSE}*\sqrt{1+Distance Value}~]\]
Confidence intervals determine the range in which the true mean lies. Whereas, the prediction interval shows the range in which the a given value could possibly fall.
We also talked about the variability of response. We covered: \(Total~Variation = Explained~Variation~+~Unexplained Variation\), or, $ = $. The percent of variables of the response that is explained by a linear relationship with the predictor can be expressed as: \(r^2~~=~~ \frac{Expected~ Variation}{Total~Variation}\).
In a simple linear regression model, r is the correlation!
Finally, we talked about the F-test. In a simple linear regression model, the F-test is the same as the t-test. However, the f-test model becomes increasingly important as we have multiple predictors.
\({H_0}:~No~aka~ \beta_1~=~0\) \({H_A}:~Yes~aka~ \beta_1~\ne~0\) The F-Test Stat is: \(\frac{Explained~Var}{(Unexplained~Var)/(n-2)}\) Under \(H_0,~~~F_{1,n-2}\)
However, it more useful just to put your data into R and let the computer do all the work!